Assalamu Alaykum Wa Rahmatullahi Wa Barakatuh
All praise and gratitude are due to Allah SWT. May the blessings and peace of Allah SWT be upon His Messenger PBUH.
This question can best be answered by referring to the events which happened during the khilaafah of Uthman RA. During his reign, he ordered for the masaahif (physical compilations of the Quran) to be burnt!
This was because a fitnah began to arise in the Muslim Ummah, where one tribe would claim that their dialect of speech is superior to another, though the general and intended meaning of speech remained constant across dialects.
Although the Quran is primarily passed on orally, it was also written in its entirety during the life of the Prophet. It was also compiled and later copied after the death of the Prophet PBUH.
Uthamn RA saw it fitting for the ummah to be united on one harf (style of reading) of the Qur’an to avoid the dispute that started to emerge and reached heightened tensions amongst the Muslims.
Uthman RA ordered for the burning of the masaahif that were not on the letter of Quraysh. Therefore, the Muslim Ummah was united on one letter, the letter of Quraysh.
It is also narrated that Uthman RA ordered for the maasahif to be cut into small pieces, or otherwise buried into the ground.
Therefore, to dispose of any material that contains the words of Allah or Allah’s name, one should not dispose of it in a place where there are impurities, as this is not befitting for the words of Allah.
The objective here is to dispose of the materials without having the words of Allah exposed to impurities. To do so, one can burn the material, bury it in a clean place that is far from the activity of people, or one can cut the material in very small pieces such that the words are no longer legible. Using a paper shredder is the most common way to dispose of papers that have the words of Allah written on it.
And Allah Knows Best.